System for formation of highly nonlinear pulses

ABSTRACT

A system supporting the formation and propagation of tunable highly nonlinear pulses using granular chains composed of non-spherical granular systems. Such a system may be used to support the creation of tunable acoustic band gaps in granular crystals formed of particles with different geometries (spherical or not) in which the tunability is achieved by varying the static precompression, type of excitation and/or pulse amplitude in the system.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a U.S. divisional application of U.S. patentapplication Ser. No. 12/364,947, filed on Feb. 3, 2009, which, in turn,is a continuation-in-part of U.S. patent application Ser. No.12/251,164, “Method and Apparatus for Nondestructive Evaluation andMonitoring of Materials and Structures,” filed on Oct. 14, 2008, and isrelated to and claims the benefit of U.S. Provisional Application No.61/063,903, titled “Method and device for actuating and sensing highlynonlinear solitary waves in surfaces, structures and materials,” filedon Feb. 7, 2008, U.S. Provisional Application No. 61/067,250, titled“System Supporting the Formation and Propagation of Tunable HighlyNonlinear Pulses, Based on Granular Chains Composed of Particles withNon Spherical Geometry,” filed on Feb. 27, 2008, U.S. ProvisionalApplication No. 61/124,920, titled “Method and Apparatus forNondestructive Evaluations and Structural Health Monitoring of Materialsand Structures,” filed on Apr. 21, 2008; the contents of all of whichare incorporated herein by reference in their entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under Grant No.CMMI0825345 awarded by the National Science Foundation. The governmenthas certain rights in the invention.

BACKGROUND

1. Field

This disclosure relates to a method and system for the formation andpropagation of highly nonlinear pulses with selectable pulse properties.More particularly, the present disclosure describes the generation andpropagation of pulses through the use of granular chains consisting ofparticles with desirable geometries.

2. Description of Related Art

The existence of the highly nonlinear regime of wave propagation insolids was discovered while studying the shock absorption properties ofgranular matter. The model typically used to represent the simplest formof granular systems consisted of a one dimensional (1-D) chain ofspherical beads regulated by Hertzian contact interaction potentials.However, a new, general wave dynamic theory, supporting compact solitarywaves, was derived for all structured homogeneous materials showing ahighly nonlinear force (F)—displacement (δ) response dictated by theintrinsically nonlinear potential of interaction between its fundamentalcomponents. This general nonlinear spring-type contact relation can beexpressed as shown below in Eq. (1):

F≅Aδ^(n)   Eq. (1)

where A is a material's parameter and n is the nonlinear exponent of thefundamental components' contact interaction (with n>1). For Hertziansystems, such as those consisting of a chain of spherical beads, the nexponent of interaction is equal to 1.5.

Within the present disclosure, “granular matter” is defined as anaggregate of “particles” in elastic contact with each other, preferablyin linear or network shaped arrangements. In addition to the nonlinearcontact interaction present in such systems, and related purely to theparticle's geometry, another unusual feature of the granular state isprovided by the zero tensile strength, which introduces an additionalnonlinearity (asymmetric potential) to the overall response. In theabsence of static precompression acting on the systems, these propertiesresult in a negligible linear range of the interaction forces betweenneighboring particles leading to a material with a characteristic soundspeed equal to zero in its uncompressed state (c₀=0): this has led tothe introduction of the concept of “sonic vacuum”. This makes the linearand weakly nonlinear continuum approaches based on Korteveg-de Vries(KdV) equation invalid and places granular materials in a special classaccording to their wave dynamics. This highly nonlinear wave theorysupports, in particular, a new type of compact highly tunable solitarywaves that have been experimentally and numerically observed in severalworks for the case of 1-D Hertzian granular systems.

SUMMARY

Embodiments of the present invention described herein include a methodand system supporting the formation and propagation of tunable highlynonlinear pulses using granular chains composed of non-sphericalgranular systems and a linearized version thereof supporting theformation of tunable acoustic band gaps. Other embodiments of thepresent invention include a method and system to support the creation oftunable acoustic band gaps in granular crystals formed of particles withdifferent geometries (spherical or not) in which the tunability isachieved by varying the static precompression, type of excitation and/orpulse amplitude in the system.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 shows a photograph of stainless steel elliptical particles.

FIG. 2A shows an experimental set up of a vertically stacked chain ofstainless steel elliptical beads.

FIG. 2B shows a sensor particle having an encapsulated piezo-sensor.

FIG. 2C shows a wall sensor having an encapsulated piezo-sensor.

FIG. 3 shows the formation of solitary waves excited by impact in achain of stainless steel elliptical beads.

FIGS. 4A and 4B illustrate relative orientations of a pair ofcylindrical particles.

FIG. 4C shows a schematic diagram of a 3-D system assembled from anarray of cylindrical contacts.

FIG. 4D illustrates the vertical alignment of the cylindrical contactsin FIG. 4C.

FIG. 5 shows experimental data obtained from a vertically aligned chainof cylinders oriented perpendicular to each other.

FIG. 6 shows a schematic diagram of a rod-based 3-D system 200 usingprecompression.

FIG. 7A shows a system in which one dimensional chains of particles areheld to each other at weld points.

FIG. 7B shows a system in which each layer is a molded layer havingindividual particles of various geometric shapes.

FIG. 8 shows a photograph of an experimental assembly used for a studyof a dimer chain consisting of alternating stainless steel and Teflonparticles.

DETAILED DESCRIPTION

Granular materials based on geometrical arrangements of spherical beadsare the simplest and most common systems used theoretically,numerically, and experimentally for studying the formation andpropagation of the highly nonlinear waves in solids. Despite being themost studied example for these systems, they are not the only onepossible solution for the creation of systems with a highly nonlinearresponse. The continuum theory derived for highly nonlinear waves indeedis not limited to the Hertzian interactions (n=3/2) between the discretecomponents: the theoretical formulation that describes them has beenextended and generalized to all nonlinear exponents n, with n≠1. Indeed,a similar power-law type response can be found in many other nonlinearsystems. The analytical formulation of the highly nonlinear waves hasalso been extended to heterogeneous systems composed of “layered”structures. Additional work may be done analytically in parallel withexperimental and numerical analysis for periodic heterogeneous nonlinearsystems. The presence of periodic “defects” (heterogeneities) isparticularly relevant for the design and study of shock protectingstructures and energy dissipaters, as the defects play a relevant rolein the scattering, redirecting sideways, or localization of energy andin the tunability of the compressive pulses traveling through thematerial. Such properties, in particular the ones found in heterogeneousgranular systems, may provide valid alternatives to the present state ofthe art shock energy protectors/dissipaters

The fundamental nonlinear dynamic response present in uniform systems isgoverned by the wave equation derived and solved in the continuum limit.For highly nonlinear uniform systems, the long wave approximation,derived from the Hertzian interaction law (n=3/2), is shown below in Eq.(2):

$\begin{matrix}{u_{tt} = {{- c^{2}}\{ {( {- u_{x}} )^{3/2} + {\frac{a^{2}}{10}\lbrack {( {- u_{x}} )^{1/4}( ( {- u_{x}} )^{5/4} )_{xx}} \rbrack}} \}_{x}}} & {{Eq}.\mspace{14mu} (2)}\end{matrix}$

where u is the displacement, a is the particle's diameter, c is amaterial's constant, and the subscripts indicate the derivative. Theconstant c in Eq. (2) is given by Eq. (3) as shown below:

$\begin{matrix}{c^{2} = \frac{2E}{\pi \; {\rho_{0}( {1 - v^{2}} )}}} & {{Eq}.\mspace{14mu} (3)}\end{matrix}$

where E is the Young's modulus, ρ₀ is the density, and v is the Poissoncoefficient. The generality of this highly nonlinear wave equation isgiven by the fact that it includes also the linear and weakly nonlinearwave equations.

Despite its apparent complexity the closed form solution of Eq. (2) canbe obtained. For the case of a granular system with no or very weakprecompression acting on it, the exact solution exists in the form asshown below in Eq. (4):

$\begin{matrix}{\xi = {( \frac{5V_{p}^{2}}{4c^{2}} )^{2}{\cos^{4}( {\frac{\sqrt{10}}{5a}x} )}}} & {{Eq}.\mspace{14mu} (4)}\end{matrix}$

where ξ represents the strain and V_(p) the system's velocity. Thesolitary shape, if the initial prestrain ξ₀ is approaching 0, can betaken as one hump of the periodic solution (provided from Eq. (4)) withfinite wave length equal to only five particle diameters.

The periodic solution described above demonstrates that in a highlynonlinear medium (such as in “granular crystals”) only two harmonicscontribute to a stationary mode of propagation of the periodic signal.In analogy with the KdV solutions (as described by Korteveg and de Vriesin “On the change of form of long waves advancing in a rectangularcanal, and on a new type of long stationary Waves,” London, Edinburghand Dublin Philosophical Magazine and Journal of Science, ser. 5, 39,pp. 422-443. (1895)), the highly nonlinear solitary waves aresupersonic, which means that their phase velocity is larger than theinitial sound velocity (c₀) in the nonlinear medium (especially in thecase of an uncompressed system, in which the c₀=0). One of their uniquefeature is the independence of their width on the amplitude (theirspatial size is always ˜5 particles diameter, no matter what waveamplitude or wave speed is present in the system) which makes them oneof the most tractable forms of “compactons” (described by Rosenau andHyman in “Compactons: Solutions with finite wavelength,” Physical ReviewLetters 70, 564 (1993)). This property is quite different from theproperties of weakly nonlinear KdV solitary waves and it is veryimportant for the use of these solitary waves as information carriersand in signal transformation devices.

The speed of the solitary wave V_(s), as a nonlinear function of themaximum particle dynamic strain in purely highly nonlinear systems, canbe expressed as shown below in Eq. (5):

$\begin{matrix}{V_{s} = {{\frac{2}{\sqrt{5}}c\; \xi_{m}^{1/4}} = {0.6802( \frac{2E}{a\; {\rho^{3/2}( {1 - v^{2}} )}} )^{1/3}\; F_{m}^{1/6}}}} & {{Eq}.\mspace{14mu} (5)}\end{matrix}$

where F_(m) is the maximum dynamic contacts force between the particlesin the discrete chain. This relationship uncovers a usefulcharacteristic of such waves, predicted by the theory and validatednumerically and experimentally: their tunability. By changing themechanical and/or the geometrical properties of the high nonlinearmedium supporting the formation of highly nonlinear solitary waves, theshape and the properties of the traveling pulse can be tuned. As such,the solitary waves in the highly nonlinear media can be engineered forspecific applications

The analytical expression for the tunability of the solitary waves speedderived from the discretization of the particles in a precompressedchain may be expressed as shown in Eq. (6) below:

$\begin{matrix}{V_{s} = {0.9314( \frac{4E^{2}F_{0}}{a^{2}\; {\rho^{3}( {1 - v^{2}} )}^{2}} )^{1/6}\; \frac{1}{( {f_{r}^{2/3} - 1} )}{\{ {\frac{4}{15}\lbrack {3 + {2f_{r}^{5/3}} - {5f_{r}^{2/3}}} \rbrack} \}^{1/2}.}}} & {{Eq}.\mspace{14mu} (6)}\end{matrix}$

where F₀ represents the static precompression added to the system,f_(r)=F_(m)/F₀ and F_(m) is the maximum contacts force between theparticles in the discrete chain. The dependence of the solitary waveproperties on the materials parameters is shown in Eq. (5) for anon-prestressed system and in Eq. (6) for a prestressed system. Anotherfeature of the highly nonlinear solitary waves is determined by the factthat the system is size independent and the solitary waves can thereforebe scalable to any dimension, according to the needs of each specificapplication. According to Eqs. (5) and (6), the tunability of the highlynonlinear solitary waves can be achieved varying one or more of thecharacteristic parameters of the nonlinear media.

The generalized form of the partial differential equation describing thehighly nonlinear regime in binary heterogeneous periodic systems hasbeen and can be expressed as shown in Eq. (7) below:

u _(ττ) =u _(x) ^(n-1) u _(xx) +Gu _(x) ^(n-3) u _(xx) ³ +Hu _(x) ^(n-2)u _(xx) u _(xxx) +Iu _(x) ^(n-1) u _(xxx)   Eq. (7)

where u is the displacement, tis a rescaled time, n is the nonlinearexponent found in Eq. (1) and the explicit expression of the parametersI, H, G can be found in Porter, M. A.; Daraio, C.; Herbold, E. B.;Szelengowicz, I.; Kevrekidis, P. G. “Highly nonlinear solitary waves inphononic crystal dimers” Physical Review E, 77, 015601(R), 2008. Asshown in Porter et al, the expressions for G, H, and I are as follows:

${G = {D^{2}\frac{( {2 - {3k} + k^{2}} )m_{1}^{2}}{6( {m_{1} + m_{2}} )^{2}}}},{H = {D^{2}\frac{2( {k - 1} )( {{2m_{1}^{2}} + {m_{1}m_{2}} - m_{2}^{2}} )}{6( {m_{1} + m_{2}} )^{2}}}},{I = {D^{2}{\frac{2( {m_{1}^{2} - {m_{1}m_{2}} + m_{2}^{2}} )}{6( {m_{1} + m_{2}} )^{2}}.}}}$

The solution for Eq. (7), describing the shape and properties of thehighly nonlinear solitary waves, from direct integration is of the formshown in Eq. (8) below:

$\begin{matrix}{{{u_{ɛ} = {v = {B\; {\cos^{\frac{2}{n - 1}}( {\beta \; \xi} )}}}},{where}}{{B = ( \frac{\mu}{\lbrack {\beta^{2}{s( {s - 1} )}} \rbrack} )^{{1/n} - 1}},{\beta = {\sqrt{\sigma}\frac{( {1 - \eta} )}{2}}}}{and}{s = {{pI}.}}} & {{Eq}.\mspace{14mu} (8)}\end{matrix}$

Highly nonlinear granular systems composed of spherical beads have beenextensively studied in the past. Embodiments of the present inventioncomprise systems that may deviate from the classical Hertzian approachassociated with systems using chains of spherical beads. Systems that donot rely upon chains of spherical beads include: chains composed ofO-rings described by Herbold and Nesterenko in “Solitary and shock wavesin discrete strongly nonlinear double power-law materials,” AppliedPhysics Letters, 90, 261902, (2007), and complex 2-D and 3-D granularassemblies as described by Goddard in “Nonlinear Elasticity andPressure-Dependent Wave Speeds in Granular Media,” Proc. R. Soc. Lond. A430, 105 (1990). Coste and Falcon describe the possibility of obtainingdeviations from the Hertzian type response also in 1-D chains ofspherical beads composed of “soft” materials (i.e. bronze or polymer) in“On the validity of Hertz contact law for granular material acoustics,”European Physical Journal B, 7, 155. (1999).

An embodiment of the present invention is a system that uses alignedstainless steel elliptical grains, such as those shown in FIG. 1. FIG. 1shows a photograph of elliptical particles fabricated from stainlesssteel. Results demonstrate that 1-D chains composed of ellipticalparticles support the formation and propagation of highly nonlinearsolitary waves when subjected to impulsive loading, following anon-Hertzian contact interaction law. FIG. 2A shows an experimental setup of a vertically stacked chain 100 of 20 stainless steel ellipticalbeads 130. Piezoelectric sensors are embedded in two sensor particles150 at particles 7 and 12, as well as at a wall sensor 170 in contactwith a wall 110. FIG. 2B shows the sensor particle 150 having apiezo-sensor 154 encapsulated in a glue layer 153 and sandwiched betweentwo particle caps 151. Particle sensor leads 152 provide an electricaloutput from the sensor particle 150. Similarly, the wall sensor 170shown in FIG. 2C has a piezo-sensor 174 encapsulated in a glue layer 173sand sandwiched between two wall sensor caps 171. Wall sensor leads 172provide an electrical output from the wall sensor 170.

FIG. 3 shows the formation of solitary waves excited by impact in thechain of twenty stainless steel elliptical beads. The twenty stainlesssteel elliptical beads (supplied by Kramer Industries) had m=0.925g±0.001 g; minor axis equal to 4.76 mm; major axis equal to 10.16 mm;modulus of elasticity equal to 193 GPa; and v equal to 0.3. The beadswere stacked in a vertical aluminum guide. Piezoelectric sensors wereprovided as shown in FIGS. 2A-2C by gluing custom micro-miniature wiring(supplied by Piezo Systems, Inc.) between the two caps of an ellipticalbead cut length-wise. The sensors were calibrated to produce forceversus time data by assuming conservation of linear momentum followingthe impact of a free falling bead. Impact was generated with 3.787 gstriker traveling at 0.75 m/s striking the top particle in the chain;the average wave speed was calculated at 525 m/s. In FIG. 3, line 191represents the data measured at the top sensor particle 150, Line 193represents the data measured at the lower sensor particle 150, and line195 represents the data measured at the wall sensor 170. It is notedthat although highly nonlinear wave theory was derived for uniformsystems with a general exponent governing their contact interaction law,experimental validation is typically provided only through Hertzianinteractions and/or using spherical particles.

According to some embodiments of the present invention, the empiricaldetermination of the “n” exponent in Eq. (1) for elliptical grains maybe determined by either of the following two methods: a first methodbased on the single particle impact and conservation of momentum; or asecond method based on the Force (F_(m))-velocity (V_(s)) scalingsimilar to that described, for example, in “On the validity of Hertzcontact law for granular material acoustics,” European Physical JournalB, 7, 155. (1999) or in Porter et al., “Highly nonlinear solitary wavesin phononic crystal dimers” Physical Review E, 77, 015601(R), 2008, fordimer chains. The second method, tested on spherical beads to verify itsrobustness has been shown to be reliable. The power law fit provided avalue of the contact interaction exponent for irregular elliptical beadsn˜1.449, proving a deviation from classical Hertzian response.

A determination of the “n” exponent from Eq. (1) for ellipticalparticles was made by analyzing the data summarized in FIG. 3 using thesecond method described above. The average velocity of the solitary wavewas determined by dividing the distance between the centers of the twoparticle sensors (equal to 5 particle diameters) by the time intervalbetween the maximum force seen at these sensors. The average maximumforce of the highly nonlinear pulses was determined by averaging theforce amplitudes at the two sensor particles. The average velocity andaverage force amplitude for the solutions generated through variousimpulsive forces provided for force versus velocity data. Evaluation ofthe power-law relationship in light of the equations above providesthat, for the measured elliptical particles, n˜1.449.

An estimate of the “n” exponent from Eq. (1) for elliptical particlesusing the first method described above was also made by impacting afixed sensor with an elliptical particle. To ensure that the particleretained proper orientation throughout free fall and contact with thesensor, a plastic guide rod was cemented to the upper portion of theparticle. Assuming conservation of linear momentum and integratingnumerically the Force versus time plots using Euler's method (beginningat the point of first contact between the elliptical particle and thesensor (t₀) until the particle reached a full stop in its descent(v(t)=0)), velocity versus time was obtained. Using the same procedure,the resulting velocity versus time curve was integrated again to producedisplacement versus time. By matching experimentally obtained force datawith calculated displacement data, a force versus displacement curve wasproduced. Best fit analysis of each resulting force versus displacementcurve enabled determination of the exponent “n. ”

Embodiments of the present invention are not limited to systems andmethods using elliptical beads. The results described earlier for 1-Dchains of elliptical beads show the formation and propagation of highlynonlinear pulses in non-Hertzian systems and support the examination anduse of 1-D granular chains composed of particles with differentgeometries. The selection of these grains having more complexnon-spherical shapes may generally require the empirical determinationof the contact interaction laws governing the Force (F_(m))-displacement(δ) response between the fundamental components of the systems; inparticular for the cases where the analytical derivation of the contactmechanics has not been provided.

Other embodiments according to the present invention include systems andmethods using particles having cylindrical geometry. One-dimensionalarrays of cylinders (as opposed to the elliptical particles describedearlier) may offer a potential for the practical assembly of 3-D systemsand enable a large range of tunability of the level of nonlinearity(value of the exponent “n” in Eq. (1)). Such tunability can be achievedby the simple variation of the reciprocal axial orientation between thecylinders in the chain as described in additional detail below.

FIGS. 4A and 4B illustrate relative orientations of a pair ofcylindrical particles. In FIG. 4A, the axis of the particles areoriented parallel to each other with θ=0°. In FIG. 4B, the axis of theparticles are oriented perpendicular to each other with θ=90°. FIG. 4Cshows a schematic diagram of a 3-D system assembled from an array ofcylindrical contacts having vertical orientations of 0°<θ<90°. FIG. 4Dillustrates the vertical alignment of the cylindrical contacts in FIG.4C.

A 1-D array of cylinders with axis oriented parallel with respect toeach other (as in FIG. 4A) do not support the formation of cleansolitary waves because of their linear contact interaction dynamics.This represents a limit case in Hertz's approach to the study ofinteraction laws between solids of revolution and presents no simpleanalytical form for its description. An axial misalignment of 0°<θ<90°where θ represents the angle between the axis of two consecutivecylinders in the chain (such as that shown in FIGS. 4C and 4D), bringsback the system to a “manageable” geometry, falling back within theHertzian treatment (n=1.5). The other limiting case (θ=90°, such as inFIG. 4B) falls back into a second limit example and does not have asimple analytical solution for the contact law.

Experimental results from a 1-D stack of cylinders oriented at 0° and90° with respect to each other has shown that by simply changing theangle of orientation between the axis of the cylinders it is possible tochange dramatically the wave propagation response of the system.Cylinders oriented at 0° (parallel axis) excited by an impulse do notshow the formation of highly nonlinear solitary waves (but ratherpresented the propagation of shock-like pulses). Chains with cylindersoriented in a 90° degrees configurations support formation andpropagation of highly nonlinear solitary pulses analogous to the oneobserved in chains of spherical beads. FIG. 5 shows experimental dataobtained from a vertically aligned chain of cylinders orientedperpendicular to each other. The chain consisted of a total of 38cylinders. Piezogauges were inserted at a wall and in 3 of the cylinderswithin the chain. The data obtained from the wall sensor is shown atline 196; the data from the cylinders within the chain are shown atlines 197, 198, and 199.

Rod-based structures similar to the one depicted in FIG. 4C can be tunedby applying variable static precompression. The application of suchstatic force can be achieved, for example, by using tension cords,strings or nets wrapped on two opposing sides of the outer cylinders orrods edges. The control over the amount of compression applied by suchelements to the assembled rods can be obtained by using smalldynamometers or by tightening screws with measured torques. FIG. 6 showsa schematic diagram of a rod-based 3-D system 200 using precompression.The system has arrays of lateral rods 201 alternating with arrays ofperpendicular rods 203. The outside ends of the lateral rods 201 arecompressed together with tension cords 210 or other such fasteningapparatus. Similarly, the outside ends of the perpendicular rods 203 arecompressed together with tension cords 210 or other such fasteningapparatus. A dynamometer 211 within each tension cord 210 may be used tomeasure and adjust the amount of precompression.

To obtain the dynamic response “purely” from the granular system withoutthe influence of the matrix between the chains, it is possible to createdesired three-dimensional systems in a cubic or hexagonal pattern atdifferent length scales assembling the particles (elliptical, conical,rods, etc) in a layer-by-layer process. The new composite granularstructures can be manufactured in large quantity in industrially viableprocesses. Depending upon the fabrication process used, it may bepossible to create light weight, tunable and even flexible or wearableprotective layers, all exploiting the new properties offered by thehighly nonlinear wave theory discussed above. Such protective systemsmay allow for sideway impulse redirection, energy trapping and/or energydissipation. FIG. 7A shows a system 230 in which one dimensional chains231 of particles 232 (that may have various geometrical shapes) are heldto each other at weld points 233 are assembled into layers for a 3-Darray of particles. Note that in FIG. 7A, the particles 232 may bewelded, glued or electrostatically/magnetically interacting together inthe horizontal direction, but are merely contacting one another in thevertical direction. FIG. 7B shows a system 240 in which each layer 241is a molded layer having individual particles 242 of various geometricshapes. Note that in FIG. 7B (similar to 7A), the layers 241 comprisemolded particles in the horizontal direction, but are the layers are 241are merely contacting one another in the vertical direction. Asdiscussed above, the systems shown in FIGS. 7A and 7B may also haveprecompression applied.

The methods and systems described above have application for acousticband gaps in tunable highly nonlinear crystals. Linear or weaklynonlinear periodic crystals with two or more atoms per primitive basis(precompressed dimer or trimer chains as described in Porter et al.,“Highly nonlinear solitary waves in phononic crystal dimers,” PhysicalReview E, 77, 015601(R), 2008. and Porter et al., “Propagation of HighlyNonlinear Solitary Waves in Phononic Crystal Dimers and Trimers,”Physica D, submitted 2007) are known to have a classical phonondispersion relation in which for each polarization mode in a givenpropagation direction, the dispersion relation develops two branches,known as the acoustical and optical branches. Depending on suchrelation, the system can present one or more frequency band gaps betweenthe branches as a function of the mass ratio in the system and theprecompression level applied to it. For a simple cubic crystal whereatoms (analogous to Hertzian grains in the systems described above) ofmass m₁ lie on one set of planes and atoms of mass m₂ lie on planesinterleaved between them, the lower bound (f₁) and upper bound (f₂) ofthe bandgap can be expressed by Eq. 9 shown below:

$\begin{matrix}{{f_{1} = {\frac{1}{2\pi}( \frac{2\beta}{m_{1}} )^{1/2}}},{f_{2} = {\frac{1}{2\pi}{( \frac{2\beta}{m_{2}} )^{1/2}.}}}} & {{Eq}.\mspace{14mu} (9)}\end{matrix}$

In Eq. 9, β is a constant proportional to the material's parameters(Youngs modulus, Poisson's coefficient and particle's radii) and staticprecompression applied to the system (see. For example, Herbold, E. B.;Kim, J.; Nesterenko, V. F.; Wang, S.; Daraio, C. “Tunable frequencyband-gap and pulse propagation in a strongly nonlinear diatomic chain”Acta Mechanica (submitted and published online), 2008).

Preliminary results were obtained from the study of dimer systems ofstainless steel and Teflon particles excited by continuous sinusoidalsignals at variable frequencies. FIG. 8 shows a photograph of theexperimental assembly used for the study in which the dimer chainconsisted of alternating stainless steel and Teflon particles. The bandgap calculated for this model system was between ˜7-14 kHz. Theexcitations with frequencies comprised in the estimated gap (as providedin Eq. (9)) remained confined in the exciter particle and its immediatesurrounding.

A numerical model for a 1-D generic granular system according toembodiments of the present invention treats particles as rigid bodiesconnected by nonlinear springs to study acoustic excitations in thesystems and the presence of band gaps, wave decay and possible presenceof gap solutions deriving from the nonlinearity of the system response.Such a model can show that when a pulse was excited within the gap, thesystem responds with a rapid decay of the initial excitation alreadywithin the first 10 particles in the chain, with relevant attenuation ofthe pulse's intensity in the audible frequency range. Thanks to the hightunability of the highly nonlinear crystals, the forbidden frequencyrange can be effectively designed and varied at will, simply choosingthe appropriate particles' mass ratio and static precompression appliedto the system.

As indicated above, embodiments of the present invention may haveparticular application to linearized granular crystals (as phononiccrystals). Just as crystalline materials can be said to possess alattice structure, with atoms occupying various positions in thelattice, phononic-crystal engineered composite systems (i.e.,“metamaterials”) can be pictured as a lattice structure with nano tomacro scale particles replacing their atomic counterparts. Such phononiccrystals based on granular materials are most fundamentally typified ina statically precompressed one dimensional (1-D) chain of macroscopicparticles. Due to zero tensile strength in the particle chain and apower-law relationship between force and displacement, linear, weaklynonlinear or highly nonlinear wave dynamics may arise, enabling theformation and propagation of solitary waves following impulsive loadingand yielding desirable properties in their acoustic and mechanicalresponse. Static compression of the particle chain prior to impulsiveloading or “pre-compression” as discussed above enables the system to betuned from highly nonlinear to weakly nonlinear to linear wave dynamics,enabling potential engineering applications in shock absorption,vibration dampening, and acoustic filtering (by forming acoustic bandgaps).

Due to the nonlinear force versus displacement relationship and thediscrete nature of granular-crystal systems, solitary (compression)waves readily form. Employing the long wave approximation, L>>a (where Lis the width of the solutions and a is distance between particlecenters), for any power law material of the form F ∞ δ^(n), the speed ofa solitary wave is given by Eq. (10) below:

$\begin{matrix}{V_{s} = {( {A_{n} \times a^{n + 1}} )^{\frac{1}{2}} \times \sqrt{\frac{2}{n + 1}}\; \times ( \xi_{m} )^{\frac{({n - 1})}{2}}}} & {{Eq}.\mspace{14mu} (10)}\end{matrix}$

where A_(n) is some constant dependant upon material properties, a isthe particle diameter (distance between two particles centers), n is theexponent governing the force versus displacement relationship, and ξ_(m)is the maximum strain in the system.

Relating ξ_(m) to the maximum force in the system (F_(m)), Eq. (10) canbe rewritten as Eq. (11) below:

$\begin{matrix}{V_{s} = {a \times \sqrt{\frac{2 \times A_{n}^{\frac{1}{n}}}{( {n + 1} ) \times m^{(\frac{n - 1}{n})}}} \times ( F_{m} )^{\frac{({n - 1})}{2n}}}} & {{Eq}.\mspace{14mu} (11)}\end{matrix}$

Just as pre-compression of a particle chain “tunes” the mechanicalresponse to impulsive loading. Eq. (11) demonstrates that adjusting theexponent (n) provides an additional means of control over linearizedgranular crystals.

From an analytical perspective, the discrete nature of a 1-D granularcrystal can be ignored if the system is treated as a continuum and ifthe particles are homogeneous in mass and material characteristics.However, the introduction of new material compositions and/or massesyields a “defect” into the system (i.e., an interface) and causes abreakdown of the analytical description of the system. Such “defects”introduce fundamentally different behavior into the granular medium andmay have potential in energy trapping/redirecting, localizationphenomena and shockwave mitigation applications. Returning to the testapparatus shown in FIG. 8, the chain of alternating stainless steel andTeflon beads demonstrated the nearly complete energy transfer across theinterface between the Teflon and stainless steel beads in theuncompressed case, as shown by the lack of a reflected compression waveinto the stainless steel beads. Introduction of multiple “defects” intoa 1-D granular crystal through an alternating pattern of differentparticles/materials like in the case of Teflon and stainless steel beadsdemonstrates the ability of such a system to transform a shock-likeimpulse into a sequence of smaller amplitude pulses.

Observation of solitary waves in a 1-D chain of elliptical beads andempirical measurement of the exponent governing the force versusdisplacement relationship for elliptical particles provides experimentalvalidation that a non-Hertzian system can support solitary wavepropagation. Such an experimental validation also demonstrates thatparticle geometry changes offer a realizable mechanism for tuning themechanical and acoustic response of linearized granular crystals.

The foregoing Detailed Description of exemplary and preferredembodiments is presented for purposes of illustration and disclosure inaccordance with the requirements of the law. It is not intended to beexhaustive nor to limit the invention to the precise form or formsdescribed, but only to enable others skilled in the art to understandhow the invention may be suited for a particular use or implementation.The possibility of modifications and variations will be apparent topractitioners skilled in the art. No limitation is intended by thedescription of exemplary embodiments which may have included tolerances,feature dimensions, specific operating conditions, engineeringspecifications, or the like, and which may vary between implementationsor with changes to the state of the art, and no limitation should beimplied therefrom. This disclosure has been made with respect to thecurrent state of the art, but also contemplates advancements and thatadaptations in the future may take into consideration of thoseadvancements, namely in accordance with the then current state of theart. It is intended that the scope of the invention be defined by theClaims as written and equivalents as applicable. Reference to a claimelement in the singular is not intended to mean “one and only one”unless explicitly so stated. Moreover, no element, component, nor methodor process step in this disclosure is intended to be dedicated to thepublic regardless of whether the element, component, or step isexplicitly recited in the Claims. No claim element herein is to beconstrued under the provisions of 35 U.S.C. Sec. 112, sixth paragraph,unless the element is expressly recited using the phrase “means for . .. ” and no method or process step herein is to be construed under thoseprovisions unless the step, or steps, are expressly recited using thephrase “comprising step(s) for . . . . ”

1.-7. (canceled)
 8. A system for the formation and/or propagation ofhighly nonlinear mechanical or acoustical pulses comprising: one or moregranular chain of particles, wherein one or more properties of at leastone particle in one or more chain of particles are chosen to providehighly nonlinear pulses having selected characteristics and whereincontact interactions between particles in the one or more granular chainof particles are non-Hertzian.
 9. The system according to claim 8wherein the one or more properties of the at least one particlecomprises a geometry of the at least one particle.
 10. The systemaccording to claim 8, further comprising apparatus applyingprecompression to at least one granular chain of particles.
 11. Thesystem according to claim 8, wherein each granular chain of particlescomprises a layer in a structure having multiple layers.
 12. The systemaccording to claim 11, wherein each layer comprises an array ofcylindrical particles of equal or different sizes and materials, whereineach cylindrical particle in the layer is arranged generally parallel toother cylindrical particles in the layer, and the cylindrical particlesin each layer are arranged generally non-parallel to cylindricalparticles in adjacent layers.
 13. The system according to claim 11wherein the particles in at least one layer are welded to, glued to orinteract by attractive forces with adjacent particles in the at leastone layer.
 14. The system according to claim 11 wherein the particles inat least one layer comprise a molded layer of particles. 15.-24.(canceled)
 25. The system of claim 8, wherein the particles in at leastone granular chain of particles are welded to, glued to or interact byelectrostatic or magnetic forces with adjacent particles.